There might be lots of different kinds
of money in there. And the other
question is about the number of
dollars…like if you counted each one.

TEACHER: Exactly. Kade is asking
about how much it’s worth, and
Keisha wants to know how many bills
are in the whole roll.

This routine of asking students
what they notice and wonder about
problems, images, and patterns was
developed by the Math Forum at the
National Council for Teachers of
Mathematics and is a helpful way to
support students in identifying the
mathematical elements of contextualized situations. If you look back at
the students’ questions, some of them
can be answered using mathematics
(How much money is there? How
many dollars are there?) whereas
others require my response (Why did
you do that? Is it your money?). Recognizing the differences between these
questions is important for students
to consider. Question formulation
and exploration are key parts of the
problem-solving process and are often
overlooked in math class.

Act 2

The majority of three-act tasks have a
focal question that is most compelling
and surfaces as the one question students want to answer. Creators of
three-act tasks often upload their Act

1 media to a site called 101 Questions( www.101qs.com) where anyone canview the media and write the firstquestion that comes to mind. Thishelps creators learn what users findmost appealing and interesting. Oncethat focal question is identified, cre-ators then work to provide sufficientresources that students will needduring Act 2 to answer that question.I had used this process for my task,and “How many bills are in the roll?”emerged as the focal question.Likewise, when I tried the task withthis class, almost all of the studentswanted to know how many bills werein the roll, so it was time for us tomove to Act 2, the richest part of theprocess. I asked the students whatinformation we’d need to answer thequestion. Josie said we needed to knowhow long the roll is, while Alli wantedto know how long a dollar is. Efranasked how they are stuck together—inother words, if they overlapped or ifthey were stuck together end-to-end.Lila offered that we needed to know ifthey were all dollar bills. When I askedwhy, she responded, “Are all bills thesame size?”The class pondered this for amoment. Some immediately said yes,but others furrowed their brows asthey thought. I gave them a moment totalk it out. One student ran to her deskand brought out a pouch where shehad a one-dollar bill and a five-dollarbill. She showed they are the samesize by holding them up together. Thismoment gave students an opportunityto work through some ambiguitytogether. In real-life problem solving,we are not given all the informationwe need in a nice, neat package—wemust gather and sort it ourselves.

In response to the students’ requests
for information, I let them know that
each bill was a one-dollar bill and that
the bills were taped end-to-end. I then
showed them an image of one dollar
with eight connecting cubes on top
(fig. 3) and an image of the roll laid
out with 179 cubes covering it (fig. 4).

Some students noticed that the
dollar was just over eight “cubes” long
because there was a little bit of space
left that wasn’t covered by cubes. This
seemed important to some students
and less so with others. Once the
students had all this information, I
invited them to work with their math
partners on the problem of how many
bills there were in all.

I walked from group to group
observing their work, listening to conversations, and at times asking probing
questions to get a sense of what
students were thinking. I refrained
from any direct teaching because the
purpose of three-act tasks is to allow
students to navigate their own solution
pathways. Some students approached
the task numerically while others
created visual representations. There
was also a range in mathematical
approaches. Some students used
division and multiplication; others
used repeated addition or subtraction.
When I encountered a group that had
an incorrect solution, I made note of
it, but didn’t intervene to help them
fix it. The emphasis of three-act tasks
is much more on the problem-solving
process than on the solution, and part
of the process is working through
what to do when you have multiple
solutions.

Act 3

The power of multiple solutions and
solution pathways is that they allow
for rich conversations during Act 3,
when students discuss and revise their
approaches. Each task is a class effort;

Question formulation and exploration arekey parts of the problem-solving processand are often overlooked in math class.